Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If ƒ is symmetric about the line 𝓍 = 2 , then ∫₀⁴ ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.

Verified step by step guidance

1. Step 1: Begin by understanding the symmetry of the function ƒ about the line 𝓍 = 2. A function is symmetric about a vertical line if, for every point (𝓍, ƒ(𝓍)) on the graph, there exists a corresponding point (4 - 𝓍, ƒ(𝓍)) that mirrors it across the line 𝓍 = 2.
2. Step 2: Use the property of symmetry to analyze the integral. If ƒ is symmetric about 𝓍 = 2, then the area under the curve from 𝓍 = 0 to 𝓍 = 2 is equal to the area under the curve from 𝓍 = 2 to 𝓍 = 4. This implies that the total integral from 𝓍 = 0 to 𝓍 = 4 can be expressed as twice the integral from 𝓍 = 0 to 𝓍 = 2.
3. Step 3: Write the integral expression mathematically. Using symmetry, we can state: ∫₀⁴ ƒ(𝓍) d𝓍 = ∫₀² ƒ(𝓍) d𝓍 + ∫₂⁴ ƒ(𝓍) d𝓍. Since the function is symmetric about 𝓍 = 2, ∫₂⁴ ƒ(𝓍) d𝓍 = ∫₀² ƒ(𝓍) d𝓍.
4. Step 4: Substitute the equality derived from symmetry into the original integral expression. This gives: ∫₀⁴ ƒ(𝓍) d𝓍 = ∫₀² ƒ(𝓍) d𝓍 + ∫₀² ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.
5. Step 5: Conclude that the statement is true based on the symmetry of the function about the line 𝓍 = 2. The integral from 𝓍 = 0 to 𝓍 = 4 is indeed twice the integral from 𝓍 = 0 to 𝓍 = 2.

Key Concepts

Symmetry in Functions

A function is symmetric about a vertical line, such as x = 2, if for every point (a, f(a)) on the graph, there is a corresponding point (4-a, f(a)). This means that the function's values are mirrored across the line x = 2, which can affect the evaluation of integrals over symmetric intervals.

Definite Integrals

A definite integral, represented as ∫ₐᵇ f(x) dx, calculates the area under the curve of the function f(x) from x = a to x = b. The properties of definite integrals, including linearity and the ability to split intervals, are crucial for evaluating integrals over symmetric intervals and understanding how symmetry impacts the integral's value.

Properties of Integrals and Symmetry

When a function is symmetric about a vertical line, the area under the curve from one side of the line can be related to the area from the other side. Specifically, if f is symmetric about x = 2, then the integral from 0 to 4 can be expressed as the sum of two equal integrals from 0 to 2, leading to the relationship ∫₀⁴ f(x) dx = 2 ∫₀² f(x) dx.